Finite Group Databasebeta

← #46839701

Subgroups — #46839701

Order 34688.

864 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864654C211yesmaximal, characteristic
2C2×#22576144C212yesmaximal, complemented
3C22×Dic271⋊C4C211yesmaximal, characteristic, complemented
4#31773818C212yesmaximal, complemented
5#31773815C212yesmaximal, complemented
6#31773774C212yesmaximal, complemented
7#31773771C214yesmaximal, complemented
8#381×C271C211yesmaximal, characteristic, complemented
9C2×#16864654—28no—
10C2×#16864654—24no—
11C22×Dic542C2211yescharacteristic
12C22×D1084C2211yescharacteristic
13C22×C22⋊D271C2212yes—
14C23×Dic271C2211yescharacteristic
15C23×Dic271C2211yescharacteristic
16C22×C4×D271C2212yes—
17#22576144—28no—
18C2×C4⋊Dic271C2214yes—
19C2×Dic271⋊C4—24no—
20C2×Dic271⋊C4C2212yes—
21C2×Dic271⋊C4C2212yes—
22C2×(C2×D271)⋊C4C2214yes—
23C2×(C2×D271)⋊C4C2214yes—
24C271⋊#95C2214yes—
25C23×C4×C271C2211yescharacteristic
26C2×C22⋊C4×C271C2212yes—
27#95×C271C2212yes—
28#95×C271C2212yes—
29C23×D271C2312yes—
30#16864654—232no—
31C2×Dic542—24no—
32C2×Dic542—22no—
33C2×D1084—24no—
34C2×D1084—22no—
35C2×C22⋊D271—24no—
36C2×C22⋊D271—24no—
37C2×C22⋊D271D418yes—
38C22×Dic271—24no—
39C22×Dic271—22no—
40C22×Dic271—24no—
41C22×Dic271—22no—
42C22×Dic271C2312yes—
43C22×Dic271C2312yes—
44C22×Dic271C2312yes—
45C2×C4×D271—24no—
46C2×C4×D271—24no—
47C2×C4×D271—24no—
48C4⋊Dic271—28no—
49Dic271⋊C4—24no—
50Dic271⋊C4—24no—
51(C2×D271)⋊C4—28no—
52(C2×D271)⋊C4—28no—
53C24×C271C2311yescharacteristic
54C22×C4×C271—24no—
55C22×C4×C271D414yes—
56C22×C4×C271C2314yes—
57C22×C4×C271C2311yescharacteristic
58C22×C4×C271C2311yescharacteristic
59C22⋊C4×C271—24no—
60C22×D271—22no—
61C22×D271—22no—
62C22×D271—24no—
63C22×D271—24no—
64C22×D271C2×D414yes—
65Dic542—28no—
66D1084—28no—
67C22⋊D271—216no—
68C2×Dic271—28no—
69C2×Dic271—22no—
70C2×Dic271—22no—
71C2×Dic271—22no—
72C2×Dic271—22no—
73C2×Dic271—24no—
74C2×Dic271—22no—
75C2×Dic271—44no—
76C2×Dic271C2×D414yes—
77C2×Dic271#4114yescomplemented
78C4×D271—216no—
79C23×C271—22no—
80C23×C271—22no—
81C23×C271C2×D414yes—
82C23×C271C2×D412yes—
83C23×C271C2411yescharacteristic
84C2×C4×C271—28no—
85C2×C4×C271—22no—
86C2×C4×C271—22no—
87C2×C4×C271—24no—
88C2×C4×C271—24no—
89C2×C4×C271—24no—
90C2×C4×C271C2×D412yes—
91C2×C4×C271C2×D412yes—
92C22×C271—22no—
93C22×C271—22no—
94C22×C271—28no—
95C22×C271#12014yes—
96C22×C271C22×D412yes—
97C22×C271C22×D411yescharacteristic
98C22×C271C2×#4112yes—
99C22×C2712−1+412yes—
100C2×D271—24no—
101C2×D271—28no—
102C2×D271—42no—
103C2×D271—42no—
104C2×D271—44no—
105C4×C271—22no—
106C4×C271—22no—
107C4×C271—44no—
108Dic271—24no—
109Dic271—24no—
110Dic271—42no—
111C2×C271—24no—
112C2×C271C2×#12011yescharacteristic
113C2×C271#52511yescharacteristic
114C2×C271#53611yescharacteristic
115C2×C271#54714yes—
116D271—44no—
117C271#240611yescharacteristic, complemented, Sylow
118#2406—2711nomaximal, Sylow
119C22×#41—2711no—
120C2×#123—2712no—
121C22×C4⋊C4—2711no—
122#395—2712no—
123#394—2712no—
124#386—2712no—
125#385—2714no—
126#381D27111yescharacteristic, complemented
127C2×#41—5428no—
128C2×#41—5424no—
129C22×Q8—2711no—
130C22×D4—2712no—
131C22×D4—2711no—
132C23×C4—2711no—
133C23×C4—2711no—
134C23×C4—2712no—
135C23×C4C2×D27111yescharacteristic
136#123—5428no—
137C2×C4⋊C4—2712no—
138C2×C4⋊C4—2712no—
139C2×C4⋊C4—2714no—
140C2×C4⋊C4—5424no—
141C2×C22⋊C4—2714no—
142C2×C22⋊C4—2714no—
143C2×C22⋊C4C2×D27112yes—
144#95—2714no—
145#95C2×D27112yes—
146#95C2×D27112yes—
147C24—2712no—
148C24C22×D27111yescharacteristic
149#41—54232no—
150C2×Q8—5424no—
151C2×Q8—5422no—
152C2×D4—2718no—
153C2×D4—5424no—
154C2×D4—5424no—
155C2×D4—5424no—
156C2×D4—5422no—
157C22×C4—24no—
158C22×C4—2712no—
159C22×C4—2712no—
160C22×C4—2712no—
161C22×C4—5424no—
162C22×C4—5424no—
163C22×C4—5424no—
164C22×C4—5422no—
165C22×C4—5424no—
166C22×C4—5424no—
167C22×C4—5422no—
168C22×C4D108414yes—
169C22×C4C22×D27114yes—
170C22×C4C22×D27111yescharacteristic
171C22×C4C22×D27111yescharacteristic
172C4⋊C4—5428no—
173C4⋊C4—5424no—
174C4⋊C4—5424no—
175C22⋊C4—24no—
176C22⋊C4—5428no—
177C22⋊C4—5428no—
178C23—22no—
179C23—22no—
180C23—2714no—
181C23—5422no—
182C23—5422no—
183C23—5424no—
184C23—5424no—
185C23D4×D27114yes—
186C23C2×D108412yes—
187C23C23×D27111yescharacteristic
188Q8—5428no—
189D4—5428no—
190D4—54216no—
191C2×C4—28no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—2714no—
198C2×C4—2714no—
199C2×C4—54216no—
200C2×C4—5428no—
201C2×C4—5422no—
202C2×C4—5422no—
203C2×C4—5422no—
204C2×C4—5422no—
205C2×C4—5424no—
206C2×C4—5422no—
207C2×C4—10844no—
208C2×C4C2×D108412yes—
209C2×C4C2×D108412yes—
210C22—22no—
211C22—28no—
212C22—22no—
213C22—5428no—
214C22—5424no—
215C22—10842no—
216C22—10844no—
217C22—10842no—
218C22#2257613014yes—
219C22C2×D4×D27112yes—
220C22C22×D108411yescharacteristic
221C22#41×D27112yes—
222C22#2257620712yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—5424no—
227C4—5424no—
228C4—10842no—
229C2—24no—
230C2—10844no—
231C2C2×#2257613011yescharacteristic
232C2#3177440911yescharacteristic
233C2#3177447311yescharacteristic
234C2#3177454514yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#381×C271C2
prime radicalC22×#16864654C2
hypercenter#381D271
hypocenterC271#2406
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #381

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#22576144C2semidirect
3. C22×Dic271⋊C4C2semidirect
4. #31773818C2semidirect
5. #31773815C2semidirect
6. #31773774C2semidirect
7. #31773771C2semidirect
8. #381×C271C2semidirect
77. C2×Dic271#41semidirect
117. C271#2406semidirect
126. #381D271semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#168646542
C2×#225761442
C22×Dic271⋊C42
#317738182
#317738152
#317737742
#317737712
#381×C2712
#2406271